[PDF] Compact planetary systems perturbed by an inclined companion: II

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Compact planetary systems perturbed by an inclined companion:

II. Stellar spin-orbit evolution

Gwenael Boue

1;2and Daniel C. Fabrycky1

boue@uchicago.edu

ABSTRACT

The stellar spin orientation relative to the orbital planes of multiplanet systems are becoming accessible to observations. Here, we analyze and classify dierent types of spin-orbit evolution in compact multiplanet systems perturbed by an inclined outer companion. Our study is based on classical secular theory, using a vectorial approach developed in a separate paper. When planet- planet perturbations are truncated at the second order in eccentricity and mutual inclination, and the planet-companion perturbations are developed at the quadrupole order, the problem becomes integrable. The motion is composed of a uniform precession of the whole system around the total angular momentum, and in the rotating frame, the evolution is periodic. Here, we focus on the relative motion associated to the oscillations of the inclination between the planet system and the outer orbit, and of the obliquities of the star with respect to the two orbital planes. The solution is obtained using a powerful geometric method. With this technique, we identify four dierent regimes characterized by the nutation amplitude of the stellar spin-axis relative to the orbital plane of the planets. In particular, the obliquity of the star reaches its maximum when the system is in the Cassini regime where planets have more angular momentum than the star, and where the precession rate of the star is similar to that of the planets induced by the companion. In that case, spin-orbit oscillations exceed twice the inclination between the planets and the companion. Even if mutual inclination is only'20, this resonant case can cause the spin-orbit angle to oscillate between perfectly aligned and retrograde values. Subject headings:methods: analytical | methods: numerical | celestial mechanics | planets and

satellites: dynamical evolution and stability | planets and satellites: general | planet-star interactions

1. Introduction

Hot or eccentric Jupiters only constitute a small

fraction of the exoplanets discovered to date.

Among those with short orbital periods, most

are smaller, less massive, and part of compact multiplanet systems (Howard et al. 2010, 2012;

Howard 2013; Petigura et al. 2013). Due to the

smaller planetary radii and larger orbital periods, the eciency of the standard method to mea-1

Department of Astronomy and Astrophysics, Univer-

sity of Chicago, 5640 South Ellis Avenue, Chicago, IL

60637, USA

2Astronomie et Systemes Dynamiques, IMCCE-CNRS

UMR 8028, Observatoire de Paris, UPMC, 77 Av. Denfert- Rochereau, 75014 Paris, France.sure spin-orbit angle in systems with hot Jupiters, based on the Rossiter-McLaughlin eect (Holt

1893; Rossiter 1924; McLaughlin 1924), decreases

signicantly in multiplanet systems. Only two multiplanet systems, called KOI-94 and Kepler-

25, have been studied with this technique (Hirano

et al. 2012; Albrecht et al. 2013). Two other meth- ods have been implemented to measure the spin- orbit angle in multiplanet systems, the stellar spot crossing technique on Kepler-30 (Sanchis-Ojeda et al. 2012), and asteroseismology on Kepler-50 and Kepler-65 (Chaplin et al. 2013). The ve systems prove to be compatible with perfect spin- orbit alignments,a priorisuggesting that multi- planet systems are preferentially in the equatorial plane of their star. A sixth system, Kepler-56

1arXiv:1405.7636v1 [astro-ph.EP] 29 May 2014

Table 1: Notation.

variable Ref. description

HamiltonianH

totEq. (33) secular Hamiltonian of the numerical system HEqs. (1, 7) secular Hamiltonian of the analytical model

KEq. (8) rst integral of the analytical problem

Eq. (35) coecients of the analytical HamiltonianH:PkandP k, respectively b (k)sLaplace coecienttimescales

1Eq. (41) precession frequency=Lofsrelative tow

2Eq. (41) precession frequency=Gofwrelative tos

3Eq. (41) precession frequency

=Gofwrelative tow0

4Eq. (41) precession frequency

=G0ofw0relative tow a,b,c,dpermutation of1,2,3,4 P nutEq. (20) nutation period P precEq. (32) precession periodstellar parametersm 0mass R

0radius

J

2Eq. (36) quadrupole gravitational harmonic

k

2second

uid Love number

Cmoment of inertia along the short axis

0rotation speed

P

0rotation period 2=!0

obliquity relative to the reference plane precession angle sstellar spin axis Lstellar angular momentumC!0sjth planet and companionm jm0mass a ja0semimajor axis b

0semiminor axisa0(1e02)1=2

P jP0revolution period e je0eccentricity I jI0absolute inclination (with respect to the reference plane) j

0longitude of the ascending node

e jeccentricity vector j jdimensionless orbital angular momentum (1e2j)1=2wj w jw0unit orbital angular momentum G jG0orbital angular momentumother variableswunit vector ofG GEq. (38) total angular momentum of the planet systemGw=PGjwj

WEq. (5) total angular momentumL+G+G0

x= cosxEq. (6) cosine of the stellar obliquity relative to the planets planesw y= cosyEq. (6) cosine of the stellar obliquity relative to the companion's orbitsw0 z= coszEq. (6) cosine of the mutual inclination between the planets and the companionww0

Eq. (17) ctitious time used to parametrize elliptic orbits in (x;y;z)geometric objectsEelliptic orbit in (x;y;z) satisfyingH(x;y;z) =handK(x;y;z) =kfor two reals (h;k)

C,@Ccube [1;1][1;1][1;1] in (x;y;z) and its boundary, respectively B,@BFig. 1 Cassini Berlingot dened byV2(x;y;z)0 and its boundary, respectively D x,Dy,DzFig. 1 diagonals which are the intersections of@Cand@B SFig. 10 hyperbolic surface equal to the union of all elliptic trajectories intersectingDx VEqs. (11, 12) oriented volume generated by (s;w;w0)

SEq. (54) quadric function dening the surfaceS

A x,AzEq. (10) length scales of the elliptic orbit in (x;y;z)Ggravitational constant cspeed of light2 (Huber et al. 2013) shows coplanarity between two transiting planets, but misalignment to the star; a distant companion, detected by radial ve- locity, may be responsible. Therefore we are led to ask what happens to the stellar obliquity if a coplanar multiplanet system is accompanied by a distant planetary companion or is embedded in a binary stellar system? Dynamically, large stellar obliquity in isolated close-in planet system can be the outcome of either planet-planet scattering (Nagasawa et al. 2008; Beauge & Nesvorny 2012) or Lidov-Kozai excitation by an outer inclined perturber (Fabrycky & Tremaine 2007; Correia et al. 2011; Naoz et al. 2011, 2012). Moreover, if the inner eccentricity is large, even a coplanar outer object can ip the planet's orbit by 180 (Li et al. 2014). More generally, in single as well as in multiplanetary systems, spin-orbit misalignment may also result from the magnetic interaction be- tween the protostar and its circumstellar disc (Lai et al. 2011) or from the solid precession of the protoplanetary disc induced by an inclined com- panion (Batygin 2012; Batygin & Adams 2013;

Lai 2014). In multiplanet systems surrounded by

an outer stellar companion, apsidal precession fre- quencies are dictated by the companionandby the planet-planet interactions. As a consequence, even at high inclination, if the planet system is suf- ciently packed, planet-planet interactions domi- nate the apsidal motion, the evolution is stabilized with respect to the Lidov-Kozai mechanism, ec- centricities remain small, and all planets move in concert (Innanen et al. 1997; Takeda et al. 2008;

Saleh & Rasio 2009). These systems are classi-

ed as dynamically rigid

1. Although the Lidov-

Kozai evolution is quenched, the planetary mean

plane still precesses if it is inclined relative to the orbit of the companion, and can eventually lead to spin-orbit misalignment with the central star. Kaib et al. (2011) applied this idea to the

55 Cancri multiplanet system which has a stel-

lar companion, and concluded that the planets are likely misaligned with respect to the stellar equator. However, the results only hold as long as the stellar spin-axis is weakly coupled to the planets orbit. We show here that this condition is not satised for the 55 Cancri system unless1 Note that our denition of dynamically rigid is more strin- gent than that of Takeda et al. (2008) who also include the

case where planet eccentricities increase in concert.the semiminor axis of the perturber is very small,

of the order of 180 au (periastron distance.30 au), whereas the projected separation is 1065 au (Mugrauer et al. 2006). Nevertheless, the required conditions for this mechanism may have been met earlier in the history of this system in particular, or other systems in general. Indeed, in our own solar system, for instance, the Sun is weakly cou- pled to the ecliptic and its obliquity of 7 might be the signature of an earlier tilt of the planet system (Tremaine 1991). Moreover, analyzing a similar problem where a protoplanetary disk takes the place of the compact planet system, Batygin (2012) showed that this mechanism is able to tilt forming planetary systems around slow rotator

T Tauri pre-main sequence stars. Here, we re-

visit the problem composed of a dynamically rigid system perturbed by a stellar or a planetary com- panion on a wide and inclined orbit. The inner planets are assumed to have low eccentricities and mutual inclinations comparable to or lower than those of our own solar system. According to these assumptions, orbital evolution induced by tides is expected to be weak and is neglected. These hypotheses are motivated by statistical studies of compact exoplanet systems detected byKepleror by radial velocity (e.g., Tremaine & Dong 2012;

Figueira et al. 2012; Fabrycky et al. 2012; Wu

& Lithwick 2013). However, we allow the overall plane of planets to tilt by an arbitrary angle. A hi- erarchical companion is included, which is allowed to have any eccentricity and inclination. The main goal of this study is to follow the evolution of the inclination of the planet system with respect to the spin-axis of the parent star. Thus, the interaction between the stellar spin-axis and the orbital mo- tion of the inner planets is taken into account. For this study, we exploit the results of the so-called \3-vector problem" which has been solved geomet- rically in (Boue & Laskar 2006, hereafter BL06) and in (Boue & Laskar 2009, hereafter BL09).

The 3-vector problem aims to model the secular

evolution of three coupled angular motions such as the lunar problem with the planet spin and the orbital angular momenta of the satellite and the star (BL06) or the binary asteroid problem with two spin-axes and their mutual orbital mo- tion (BL09). Here, the three vectors are the spin of the star, the total orbital angular momentum of the planet system, and that of the companion. 3 In Section 2, we recall the main results of the 3- vector problem, and we also provide a new integral expression for the precession frequency. Then, in Section 3 we employ the vectorial formalism of the classical secular theory that we described in a pre- vious paper (Boue and Fabrycky 2014; BF14) and we show how the three vector problem emerges from this general secular model. The validity of the simplication is also discussed. In this work, we thus consider two dierent models which cor- respond to two levels of approximation. On the one hand, the perturbing function is expanded at the fourth order in planet eccentricity and mu- tual inclination and at the octupole in the inter- action between each planet and the companion.

This model provides accurate results but is non-

integrable and has to be solved numerically. On the other hand, the system is described by the in- tegrable three vector problem which gives deeper geometrical insight. In the following, we refer to the former as thenumericalmodel and to the lat- ter as theanalyticalmodel. In Section 4, the two models are compared in their application to real exoplanet systems. Then, we exploit more deeply the possibilities of the analytical model to span the parameter space and identify four dierent regimes of evolution in Section 5. The conclusions are given in the last section.

2. Three-vector problem

This section summarizes a few key results as-

sociated to the so-called \3-vector problem" de- scribed in (BL06; BL09). In the context of this paper, the three vectors are the angular momen- tum of the starL=Ls, the orbital angular mo- menta of the planet systemG=Gw, and that of the companionG0=G0w0, wheres,w, andw0 are unit vectors. The 3-vector problem assumes that the evolution is governed by a Hamiltonian of the form H=2 (sw)22 (sw0)2 2 (ww0)2;(1) where,, and are constant parameters repre- senting the coupling between the planetary system and both the stellar rotation and the binary orbit, respectively. Their expression will be derived in the subsequent section. Note that in contrast to the more general 3-vector problem, here we neglect the direct interaction between the stellar spin andthe orbit of the companion, i.e., we set= 0. The equations of motion are dsdt =1L srsH; dwdt =1G wrwH; dw0dt =1G

0w0rw0H ;(2)

which leads to dsdt =L (sw)ws; dwdt =G (sw)sw G (w0w)w0w; dw0dt G

0(w0w)ww0:

(3)

In prevision of the subsequent analysis, we set

1==L ;

2==G ;

3= =G ; 4= =G0:(4)

These quantities are important as they are the

characteristic precession frequencies ofsaround w, ofwaroundsandw0, and ofw0aroundw, respectively.

2.1. Integrability

The 3-vector problem is integrable (BL06;

BL09). Let

W=Ls+Gw+G0w0(5)

be the total angular momentum of the system.

The general solution is a uniform rotation of the

three vectors around the total angular momentum combined with a periodic motion in the rotating frame (BL06; BL09). The evolution is thus char- acterized by two frequencies or periods. Hereafter, the uniform rotation is referred to as the preces- sion motion with periodPprec, and the periodic loops described in the rotating frame are equally qualied as nutation in reference to the Earth-

Moon problem, or simply as the relative motion

with periodPnut. The relative motion can be solved elegantly with geometric arguments (BL06; 4 BL09). It is also very important for our study for two reasons: it enables 1) to check if any system can be misaligned, and 2) to evaluate the timescale of the secular spin-orbit evolution which can then be compared to the lifetime of the system. Next, we recall its solution and main properties as de- rived in (BL06; BL09). Then, we present a new integral expression of the precession period.

2.2. Relative motion

In order to get the relative evolution of the sys- tem described by the Hamiltonian (1), we follow the same derivation as in BL06 and BL09. We denote x=sw; y=sw0; z=ww0:(6)

Sometimes, we will also use the corresponding

angles dened byx= cosx,y= cosy, and z= cosz. In this coordinate system, the Hamil- tonian reads as H=2 x2 2 z2:(7)

The conservation of the norm of each angular mo-

mentumL,G, andG0, as well as the total angular angular momentum of the systemW, Eq. (5), lead to the second constant of the motion

K=kWk2L2G2G022

=LGx+LG0y+GG0z :(8)

Each trajectory of the relative motion in the

(x;y;z) frame is at the intersection of a cylin- der dened byH(x;y;z) =hand a plane dened byK(x;y;z) =k, wherehandkare two con- stants given by the initial conditions. Trajectories are thus subsets of ellipses dened by the values handkof the two rst integrals of the motion.

Hereafter, we denote them asE=f(x;y;z)2R3j

H(x;y;z) =h;K(x;y;z) =kg. These ellipses can

be parametrized as follows x() =Axcos ; z() =Azsin ; y() =1LG

0kLGx()GG0z();(9)where

A x=r2h ; Az=s2h :(10)

The change of timet7!leading to the

parametrization (9) will be made explicit in sec- tion 2.4. In general, systems do not cover the full ellipses. Indeed,x,y, andzare dot products of unit vectors and the evolution is restricted inside the cubeC=f(x;y;z)2[1;1]3g. There is also a more stringent additional constraint (BL06). Let

V=s(ww0):(11)

V represents the oriented volume of the paral-

lelepiped generated by the vectorss,w, andw0.

In terms of the dot productsx,y,z, the square of

the volumeVis given by the Gram determinant V 2= 1x y x1z y z1 = 1x2y2z2+2xyz :(12)

When the three vectorss,w, andw0are copla-

nar,V2(x;y;z) = 0. This is the equation of a cubic surface known as Cayley's nodal cubic.

The restriction of this surface to the cubeCis

displayed in Fig. 1. In BL09, this restriction is calledCassini Berlingot. The word Berlingot comes after a french hard candy with a similar shape. The name Cassini has been added in ref- erence to the `Cassini states' characterized by the coplanarity of the same three vectors as in this problem. Thus `Cassini states' are located at the surfaceV2(x;y;z) = 0. Because the cubic (12) represents the square of the volumeV, it must be positive. As a consequence, the evolu- tion of the system is restricted inside the Berlin- gotB=f(x;y;z)2 C jV2(x;y;z)0g. Here,

Bdenotes the inside of the Berlingot, and@B

its surface. In Fig. 1, important diagonalsDx, D y, andDzare represented. They all belong to the intersection@B \@C, in which the three vec- tors (s;w;w0) lie in the same plane. The point (1;1;1) corresponds to the conguration where all three vectors are aligned and in the same direc- tion. In that case, the system is fully coplanar with only prograde orbits. At the point (1;1;1), the three vectors are still collinear, butw0is pointing in the opposite direction assandw. In that case, 5 D xD zD yFig. 1.| Cassini Berlingot dened by V

2(x;y;z)0. AsV2must be greater or

equal to zero, the allowed region in the (x;y;z) space is the interior of the Berlingot shape volume. The diagonalsDx,Dy, andDzare also represented. See text for detail. the system is also coplanar, but the outer compan- ion is on a retrograde orbit. Along the diagonal Dquotesdbs_dbs27.pdfusesText_33

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